Question

If f : C → C is defined by f(x) = x⁴, write f⁻¹ (1).

Answer 1

$$\begin{gathered} \text{Let}{f}^{-1}\left(1\right)=x...\left(1\right) \\ \Rightarrow f\left(x\right)=1 \\ \Rightarrow x^4=1 \\ \Rightarrow x^4-1=0 \\ \Rightarrow \left(x^2-1\right)\left(x^2+1\right)=0\left[\text{u}\mathrm{sing}\mathrm{identity}:a^2-b^2=\left(a-b\right)\left(a+b\right)\right] \\ \Rightarrow \left(x-1\right)\left(x+1\right)\left(x-i\right)\left(x+i\right)=0,\text{w}\mathrm{here}i=\sqrt{-1}\left[\text{u}\mathrm{sing}\mathrm{identity}:a^2-b^2=\left(a-b\right)\left(a+b\right)\right] \\ \Rightarrow x=\pm 1,\pm i \\ \Rightarrow f^{-1}\left(1\right)=\left\{-1,1,i,-i\right\}[\text{from}\left(1\right)\text{]} \end{gathered}$$